This invention relates to an amusement device, more particularly to a puzzle device.
Manipulative puzzles whose object is to arrange game pieces into a particular order have enjoyed popularity for centuries. Similar to the current invention are "Shunting Puzzles". Puzzles of this type can be generally characterized as having a series of game pieces arranged in a linear fashion, with some visual characteristic for distinguishing one game piece from another, arranged on a track so that the game pieces may slide along the track. The track is designed in such a way that game pieces may be moved to change the relative relationships with each other, thereby accomplishing the object of these puzzles.
U.S. Pat. No. 4,871,173 by Lammertink, U.S. Pat. No. 4,832,343 by Bernat, U.S. Pat. No. 5,110,130 by Auliclno and U.S. Pat. No. 5,114,148 by Liu describe closed tracks comprising track segments on rigid bodies. U.S. Pat. No. 4,767,120 by Ho, U.S. Pat. No. 5,299,804 by Stevens, U.S. Pat. No. 3,815,280 by Gilfillan, and Aulicino just mentioned describe connection of rigid bodies via pivots in a continuous loop, although none of these specifies six bodies in a loop. Of those mentioned, the author considers Bernat to be closest in function to the present invention.
Bernat describes a series of track segments whose movements are confined by way of connecting pieces, but which can be moved into various positions in which the track segments combine to make a closed track. Track segments are completely filled with game pieces, but may be moved simultaneously along any such closed track. The Bernat puzzle is solved by alternating movement of track segments to form closed tracks with movement of game pieces within closed tracks thus formed.
The advantage of the current device over other mechanical puzzles is not in the difficulty of finding a solution, but in the fascination kindled by 3-dimensional movement of a structure that at first appears immovable. This movement is the feature which makes the current device unique.
Basic Concept
The designing of embodiments of this device other than the one described below is based on an understanding of why a symmetrical pattern of six pivots connecting consecutive bodies in a loop is advantageous. A discussion of theory follows.
Mentally assembling a device while cautiously counting the number of dimensions of movement gained and lost when each piece is introduced can reveal mobility, or lack thereof, within that device. Caution is advised because sometimes a restriction seen by mathematical analysis allows movement only in an imaginary direction, which translates into immobility. On the other hand, a limitation of movement which is redundant will eliminate less movement than expected. There is redundancy in the current device in a way which is not obvious. The following hypothetical assembly without imposing symmetry will suggest immobility of the axes. A second assembly with symmetry imposed will allow movement.
For the first assembly, symmetry is not imposed and it is assumed that movement restrictions are independent, i.e. not redundant. Placement of a line in space has four dimensions of movement. I.e. within any suitable fixed frame of reference, four numbers would be required to uniquely express its position and orientation. Independent movement of six such lines would thus have twenty-four dimensions of movement. Whenever a moveable but rigid body is introduced and two of these lines are held rigid with respect to this body, both the shortest distance between the two lines and the angle between the directions of them become fixed. These two restrictions being given and constant, two of the total dimensions can be determined by all others, so that the dimension count will decrease by two. Six such rigid bodies are introduced, so as to involve said lines and rigid bodies alternately in a loop. This decreases the original dimension count of twenty-four by twelve (2.times.6), leaving twelve dimensions of movement. Whenever any two rigid bodies that involve a common line are then actually connected with a pivot, that line becomes a pivot axis. Translation of one body with respect to the other is denied, thus subtracting one dimension. Six such pivots are formed, thus subtracting six from the twelve dimensions present at this point, leaving six. Since there is no stationary connection of any part to the frame of reference, six dimensions are necessary to locate and orient the entire group within space. If six dimensions of movement are remaining and six are required to locate and orient the agglomeration of bodies, then movement within the agglomeration is not possible.
One may generalize, and consider n bodies connected in similar fashion. The number of dimensions of freedom is similarly anticipated to be n-6 when n is 6 or greater. As mentioned above, movement might be mathematically possible only in an imaginary direction, as when each of eight rigid bodies hold consecutive axes to intersect at 135 degrees, resulting in an immobile octagon of axes.
Returning to the original case of six axes for the second mental assembly, a line of symmetry is imposed. This line of symmetry has four dimensions of movement. Likewise, proposing a first pivot axis anywhere in space adds four dimensions, but the position of its symmetrical counterpart is thereby determined. Thus, placing pivot axes while imposing symmetry but not yet considering which lines are connected with rigid bodies, the axes will add twelve dimensions in place of the twenty-four involved in the first mental assembly. The sum is now sixteen. Similarly, every restriction made between parts in one place will impose the same restriction between symmetrical counterparts, thus cutting in half the number of dimensions lost. Instead of subtracting twelve when rigid bodies are introduced, only six are subtracted, to leave ten. Instead of subtracting six when pivots are put in place, three are subtracted. The result is seven, in place of the six in the first assembly. Allowing the six dimensions of location and orientation of the group, this allows one dimension of movement within the group.
Line symmetry is thus one way that exactly six axes may be made to be moveable. If we generalize the above, the indication is that a loop of n rigid bodies, where n must now be even, built with bodies to allow line symmetry and confined to that symmetry by whatever means during movement, we find that n/2-2 dimensions of movement should exist. For eight bodies, the original n-6=2 dimensions are retained. Above eight, symmetry is confining rather than liberating. Below six, line symmetry by itself offers no additional dimensionality. Therefore, a figure of six axes is unique in this regard.
Without imposing symmetry, a loop of seven rigid bodies in general has one dimension of movement; however, considering small movements to or from any specific orientation, some pivots experience larger movements compared to those on the other side of the loop. Designing track segments that will align is a far more challenging exercise than it is with six. Use of six rigid bodies causes the pivot action to be more evenly distributed around the loop.